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The cold-to-warm joint is often used in the cryogenic system to take advantages of both its thermal and mechanical performances. In this study, we report the experimental measurement of the thermal conductivity and thermal conductance of the cold-to-warm joint for the magnet using from cryogenic temperature to room temperature. The cold-to-warm joint is composed of the carbon fiber reinforced plastics (CFRP) cylinder and metal parts. Two kinds of CFRPs are made of the T300 and T700 carbon fibers and epoxy, respectively. The thermal conductivity of CFRP is measured by a 4 K G-M cryo-cooler based experimental facility and it is measured to be about 0.15-5.0 W/(m·K) in about 40 K to room temperature. And the thermal conductance of the cold-to-warm joint is also estimated based on the experimental data. Those results are essential information for designing the suspension structure in cryogenic applications.